Learn English – coming to the shops or going to the shops?

They're both correct, and the difference between the two is subtle.

To my mind, as a native speaker of British English, if I ask

Do you mind coming to the shops with me?

...it would imply that I will go to the shops, regardless of whether you accompany me or not. In a sense, I would be 'leading' our expedition, and we would be visiting these emporia chiefly for my own purposes. However, asking

Do you mind going to the shops with me?

...connotes that my travelling to the shops is contingent on your coming with me, or at the very least, I have yet to make up mind.

As I said, though, this really is a point of nuance in meaning, and as such it is likely to differ between regions, dialects and quite probably individuals.